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An experimental and numerical study of bimodal velocity profile of longshore currents over mild-slope barred beaches
Ocean Engineering 106 (2015) 415–423
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An experimental and numerical study of bimodal velocity profile of longshore currents over mild-slope barred beaches Yan Wang n, Zhili Zou State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China
art ic l e i nf o
a b s t r a c t
Article history: Received 24 October 2014 Accepted 24 June 2015
A detailed description is given of the results of laboratory experiments on bimodal velocity profile of longshore currents over two mild-slope barred beaches. The objective is to examine the bimodal longshore current velocity profile for purely wave-driven currents, with emphasis on the second peak and ratio of two peaks. The research focus is on the locations, values and ratios of two velocity peaks for the cases of mild slope beaches. The dependence of its bimodal feature on wave height, wave period and beach slope on the longshore currents are discussed. The 2-D model based on vertically integrated equations (external mode) of Nearshore POM was performed to compute the measured velocity profile. The depth-varying wave-induced residual momentum, the surface rollers, and the lateral mixing the bottom friction are considered, and the effects of lateral mixing, surface rollers and bottom friction on numerical model are discussed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Mild slope Barred beach Bimodal feature Second peak POM model Wave roller
1. Introduction When water wave on beach propagates obliquely to the shoreline and break, a mean current is generated flowing parallel to the coast. For alongshore uniform barred beaches, the distribution of longshore current usually has two peaks, each corresponding to one of wave breaking points. The maximum peak of longshore current is expected to occur at the point of most intense wave breaking (Ruessink et al.,2001; Feddersen and Guza, 2003), i.e. on the bar crest. The minimum peak is expected to be close to the shore line. But in the case of field measurement, it is shown that a maximum longshore current velocity happens in the bar trough, instead of bar crest, as observed during the DELILAH field experiment results at Duck, North Carolina and the PNEC field experiment results on French Aquitanian coast beaches. Laboratory experiments on the shear instability of longshore currents have been carried out (Reniers and Battjes, 1997; Ren et al., 2011). This is contrary to the corresponding laboratory experiments and numerical simulation (Reniers and Battjes, 1997; Ruessink et al., 2001), which give the maximum velocity peak near the bar crest. Several mechanisms have been examined to explain this difference, including wave breaking turbulence, rollers, shear instabilities and alongshore pressure gradients (Putrevu and Svendsen, 1994). But these studies are most for relatively steep beach. The present study examines this problem for a mild slope case.
n
Corresponding author. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.oceaneng.2015.06.038 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
The related experiment study for steep beach (1:20) was carried out (Reniers and Battjes, 1997). They used an external circulation system to maintain the flow circulation on beach caused by longshore current, which is similar to that of Visser (1984) for the longshore current on a plane beach. Their results show that the maximum velocity peak is on the bar crest. This result is also confirmed by their numerical simulation. The second velocity peak is not examined in detail in this study, because not enough velocity meters were adopted, and the exact location of velocity peak cannot be measured. Goda (2006) and Ruessink et al. (2001) also performed similar numerical simulations to investigate the cross-shore longshore current distributions on barred beach and similar results were obtained. Choi and Yoon (2011) and Choi et al. (2012) improved the wave-breaking model of Battjes and Janssen (1978) and the surface roller model of Tajima and Madsen (2006) for the application of SWAN-SHORECRC into the Sandy Duck experiments and its application showed satisfactory performances. Newberger and Allen (2007a, 2007b) added wave forcing in the form of surface stress and body forces in the Princeton Ocean Model (POM), which has evolved as “Nearshore POM”. Nayak et al. (2012) simulated the nearshore wave-induced setup along Kalpakkam coast during an extreme cyclone event in the Bay of Bengal using the SWAN wave model and the vertically integrated momentum balance equation in which a wave-induced setup is balanced by the wave radiation stresses. The present study investigates the above problem experimentally for the two mild slope cases and more velocity meters are adopt in all to measure the velocity peak, especial minimum peak.
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The numerical simulations are also conducted to explain the measurement. The purpose of the present paper is to investigate the bimodal longshore current velocity profile for purely wavedriven currents, with emphasis on the second peak and ratio of two peaks. In the present study, the longshore currents produced by regular waves are measured on the two barred beaches with slopes of 1:100 and 1:40. In view of the discussion on the position of the two peaks of longshore current in relation to the areas where wave energy is being dissipated by two wave breaking, we feel that the measurements on the cross-shore distribution of the longshore current velocity are valuable enough to be presented separately. The results can be used to obtain a better understanding of the forcing of wave-driven currents in the field. Experiment techniques of passive recirculation system are studied. The computational methods of longshore currents are investigated for simulation of the experimental results of longshore currents and wave heights for regular wave cases. The organization of the paper is as follows. Following the introduction, Section 2 presents the experiment setup. Section 3 shows the alongshore uniformity. Section 4 discusses the bimodal features of longshore currents. Section 5 establishes the numerical model with the effects of lateral mixing, bottom friction and surface rollers to compute the measured velocity profile and discusses the effect of lateral mixing, surface rollers and bottom shear stress on numerical results.
Fig. 1. Experimental setup (top) and layout of ADV (bottom). Measurement in meters.
2. Experiment setup The measurements were performed in the wave basin of the State Key Laboratory of Coastal and Offshore Engineering in Dalian University of Technology, which is 55 m long, 34 m wide and 0.7 m deep. Fig. 1 shows the experimental setup. In order to examine the bimodal feature of longshore current over bared beach with mild slope, 1:100 and 1:40 slope beaches were made by concrete. In order to form an obliquely incident wave on the beach and increase the beach length, the beach was rotated at 30o with respect to the wave-maker, which is at the one end of the wave basin. At each end of the beach a channel was made to maintain water circulation caused by longshore currents. The channels around the beach topography had the same depth as that in the flat part in front of the beach. Two wave guide walls were conducted on the side boundaries on the horizontal bottom part of the flow region. The wave absorbing layers formed by net boxes filled with plastic scraps were placed on the inner side of to prevent the wave reflection against the walls. Fig. 2 shows experimental photo of slope 1:100. The coordinate system (x, y) with the origin at the upstream end of still water shoreline was adopted with x-axis directing offshore and y-axis downstream. The measurements were performed on two barred concrete mild slopes with 1:40 and 1:100 slopes (Fig. 1) on which a Gaussian bar profile with a crest height of about 0.08 m or 0.04 m, width of 2.0 m and located at x¼ 5.0 m or 7.0 m was superimposed. The Gaussian bar profile was selected to reflect conditions observed in nature, with intense wave breaking over the bar, followed by waves reforming in the trough and additional wave breaking near the shoreline. The wave surface elevations were measured by 60 capacitance wave gauges placed in three measuring sections (I–III), which were normal to shoreline and have equal interval of 5 m. The distance from the upper end of the beach to measuring section I is y¼7 m. The measuring sections I and III both have 14 gauges with interval 2 m. The measuring section II has more wave gauges (32) and small gauge interval (0.5 m from 0.5 m to 10 m and 1 m from 10 m to 22 m) in order to measure the location of breaking point more accurately. The measurement results of these three
Fig. 2. Photograph of the experiment.
measuring lines were used to check the longshore uniformity of wave height, wave setup and set down. A total of 28 velocity meters (ADVs) were used for velocity measurement. They are divided into two sets, one being arranged in the longshore direction to measure the uniformity of longshore current (located at x ¼4 m (slope 1:100) and x ¼3 m (slope 1:40) with interval 2 m); the other being arranged in the cross-shore direction to measure the cross-shore distributions (located at y¼14.5 m intervals 0.5–2.0 m). The vertical distance from bottom of each ADV is 1/3 local water depth for the measurement of depth-average velocity. The sample length of ADVs recording was 600 s. Fig. 3 shows time series of 5 flow meters in section IV in Fig. 1, from which we can see that the recording of every flow meter has a steady mean value after 200 s, indicating that the steady longshore current is achieved. We also see the fluctuation of the longshore current from the figure, and the period of fluctuation is much longer than the wave period (wave period is 1.0 s). This shows that the flow meters cannot respond to wave velocity simultaneously and only give the results of wave averaged velocity. The steady velocity data cut from about 300 s in the time series were used to determine the mean longshore currents. The mean longshore velocity is obtained by time-averaging of the recorded velocity time series cut by about 300 s from the time series beginning. Unidirectional obliquely incident regular and irregular waves were generated by the multi-paddle-type wave maker. Table 1 lists
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the incident wave conditions, where T is the wave period, H 0 is mean wave height for regular wave and the significant wave height, h is the still water depth over horizontal bottom.
3. Alongshore uniformity Different from the previous researches, a passive, instead of active, circulation system is adopt to form flow circulation caused by the longshore currents, as the active circulation system (Reniers and Battjes, 1997) needs a manmade flow flux in the up and down stream ends of the beach which is difficult to determined accurately and will affect the accuracy of measured longshore current. The passive circulation system employed in the present study was formed by the channel around the beach model, as shown in Fig. 1 by arrows. The channel width is 4.4 m at the two sides of the beach and 4.0–8.0 m behind the beach. The channel water depth is the same as that in the horizontal bottom in front of the beach. The flow in the downstream channel will be driven by the longshore currents and goes through the channel behind the beach model to reach the upstream channel. Then, it feeds into the upstream end to form a closed flow circulation. The longshore current formed by this circulation system has a small longshore velocity at the upstream end (because the driving force (radiation
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stress) does not get properly formed due to the lateral boundary effect) and grows linearly to almost uniformed longshore velocity over a milder part of the beach, and then have gradual increase toward the downstream end (because of the same reason as at the upstream end). This form of a passive circulation system is similar to that discussed theoretically and experimentally by Dalrymple et al. (1977), which has a closed upstream end but an open downstream end. The theoretical and experimental streamline of such a system were given and it is seen that uniformly distributed streamline are formed over the middle and downstream part of the beach. The similar longshore flow pattern was also found in the present experiment, as shown in Fig. 4. This is especially true for a short wave period case (T¼ 1.0 s) (see the discussion of following subsection). The longshore uniformity of longshore currents generated by the passive circulations system is examined by the mean longshore current recorded by the alongshore set of ADVs. Fig. 4 shows such longshore currents of different wave height for each wave period considered, in which results along the measuring line, x¼3 m for slope 1:40 and x¼ 4 m for slope 1:100 are presented. Another wave period (1.5 s and 2.0 s) for each of the last 4 panels in the figure (which is not listed in Table 1 as it is only considered here) is added in order to consider the wave conditions for three wave periods. It can be seen for the cases with T¼1.0 s and slopes 1:40, 1:100, and the cases with T¼1.5 s and slope 1:40 that the longshore currents is uniform from y¼8.5 m to 18.5 m, and the longshore currents velocity in the upstream is smaller and the longshore currents velocity in the downstream is larger. For the cases with T¼ 2.0 s and slope 1:40, the longshore currents velocity becomes significantly increased from y¼ 12.5 m to 22.5 m and the region of longshore uniformity gets shorter. For the cases with T¼1.5 s or 2.0 s and slope 1:100, the apparent fluctuations are shown in longshore direction. Therefore it was found that, with the increased wave period, the longshore uniformity of longshore current get worse and the region of uniform longshore current becoming shorter. The cases with T¼1.0 s and slopes 1:40, 1:100, and the cases with T¼ 1.5 s, 2.0 s and slope 1:40, the length of the uniform longshore current is long (one third of total beach length) enough for the measurement of the longshore current velocity profile. The longshore uniformity of the wave height and wave setup was also checked by the surface elevation measurement in measuring sections I, II and III. Fig. 5 shows the measured wave height and setup for test 1 (slope 1:40) and test 7 (slope 1:100) over barred beaches. As shown in figure, the wave height and wave setup are approximately uniform in the basin for barred beaches, which form mean longshore current. This is important in view of the question raised in Section 1, since it implies absence of an alongshore pressure gradient and a purely local wave-induced driving force. Three runs for each test condition were conducted and it is found that the repeatability of these results is quite good.
4. Bimodal features of cross-shore distribution of longshore currents In this section the factors influencing the cross-shore distribution of measured longshore currents are presented, which include the wave height, wave period and bottom topography, whose
Fig. 3. The time series of longshore current velocity (test 2). Table 1 Wave conditions. Test
1
2
3
4
5
6
7
8
9
10
Slope d/m T/s H 0 /m
1:40 0.45 1.0 0.070
1:40 0.45 1.0 0.090
1:40 0.45 1.0 0.12
1:40 0.45 1.5 0.070
1:40 0.45 2.0 0.070
1:100 0.18 1.0 0.025
1:100 0.18 1.0 0.050
1:100 0.18 1.0 0.060
1:100 0.18 1.5 0.050
1:100 0.18 2.0 0.050
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Fig. 4. A longshore distribution of mean longshore current.
Fig. 5. Longshore uniformity of wave height and setup for barred beaches (tests 1, 7).
effect is given in details in the next subsection. Fig. 6 shows the cross-shore profiles of wave height, setup and longshore current velocity, which correspond to different wave heights of T ¼1.0 s shown in Table 1. The measurement data of wave height and setup is from the measuring section II in Fig. 1. The left panel is for the cases of slope 1:100 and the right panel for the cases of slope 1:40.
The main feature of the velocity profile which is different from the plane beach is bimodal: there are two velocity peaks corresponding to two wave breakings. Table 2 summarizes the locations, values and the ratios of these two peaks for same period (1.0 s) but different wave height. The following gives the details discussion of the results.
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Fig. 6. Cross-shore profiles of wave height, setup and longshore current velocity of T ¼1.0 s.
Table 2 First and second velocity peaks for same period (1.0 s) but different wave height. Test xð1Þ =xbar xð2Þ =xbar First peak b b Value/ m s1
Location
xð1Þ c =xbar
xð2Þ c =xbar
Value/ m s1
0.90 0.90 0.90 0.93 0.93 0.93
0.276 0.369 0.421 0.067 0.146 0.153
0.25 0.25 0.25 0.21 0.21 0.21
0.116 0.153 0.169 0.018 0.040 0.044
Location
1 2 3 6 7 8
1.1 1.3 1.6 1.07 1.43 1.71
0.30 0.30 0.40 0.29 0.29 0.29
Second peak
Ratio of two peaks
0.42 0.41 0.40 0.27 0.27 0.28
ð2Þ xð1Þ ; xð2Þ : locations of two wave breaking points; xð1Þ c ; xc : locations of two peaks of b b longshore currents.
4.1. Locations and values of two peaks It is seen from the wave height distributions in the figures that with wave propagating toward shoreline the wave breaking happens two times for the barred beaches. The first wave breaking occur mostly on offshore sides of sand bar, because the water depth around bar crest is 0.03 m (slope 1:100) and 0.045 m (slope 1:40), which is so small that the breaking condition H 4 γ h can be satisfied easily on offshore sides of sand bar. The second wave breaking occurs near the shoreline. Corresponding, there are two peaks of longshore current velocity profile. As the first peak of longshore current corresponds to the first wave breaking, it is controlled by the sand bar. But there is a distance lag between
velocity peak and breaking point, as shown in Table 2 by the values of xbð1Þ =xbar and xð1Þ c =xbar : the difference between two values is the distance lag. For the second velocity peak and second wave breaking point, this lag also exist, which is shown by the values of xð2Þ =xbar and xð2Þ c =xbar in Table 2. These results show that the first b peak of longshore current velocity all occurs at the onshore side of ð1Þ sand bar: xð1Þ c =xbar ¼ 0:90 for slope 1:40 and xc =xbar ¼ 0:93 for slope 1:100. This result is a little bit difference from the experimental result (xcð1Þ =xbar ¼ 1:0) by Reniers and Battjes (1997). This may be because there was not measurement points between the crest and trough of the bar in their experiment, so the exact location of velocity peak cannot be determined for their experiment. Both the present and Reniers and Battjes's results are different from the field experiments of DELILAH (1991) and PNEC (2006), which indicate that the first peak of longshore current velocity is located in the bar trough. The second peak of longshore current all occurs near the shoreline, x¼1.5 m (corresponding to ð2Þ xð2Þ c =xbar ¼ 0:21) and x ¼1.25 m (corresponding to xc =xbar ¼ 0:25), where the remaining wave energy was dissipated after propagating over the bar. Table 3 gives the ratio of second peak to first peak for different cases, which are near 0.41 (slope 1:40) and 0.27 (slope 1:100). It is interesting to notice that for each slope, the locations and ratio of the two peaks in the table remain unchanged for different wave heights and wave types. 4.2. Effect of wave height The wave height effect is studied in the present experiment by varying the wave heights of incident wave for a fixed wave period,
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Table 3 First and second velocity peaks for similar wave height but different period Test T/s First peak
1.0 1.5 2.0 1.0 1.5 2.0
Ratio of two Figure peaks
Value/ m s1
Location
xð1Þ c =xbar
xð2Þ c =xbar
Value/ m s1
0.90 0.90 0.90 0.93 0.93 0.93
0.276 0.275 0.301 0.067 0.151 0.152
0.25 0.25 0.25 0.21 0.21 0.21
0.42 0.55 0.59 0.27 0.31 0.39
Location
1 4 5 6 9 10
Second peak
0.116 0.151 0.179 0.018 0.046 0.059
Fig. 7 (a)–(c) Fig. 7 (d)–(f)
ð2Þ xð1Þ c ; xc : locations of two peaks of longshore currents.
which are from 0.07 m to 0.12 cm for slope 1:40 and 0.025 m to 0.060 cm for slope 1:100. It is seen in Fig. 4 that the increasing of wave height results in wave breaking starting further offshore, and this leads to a broader breaking zone and a broader current velocity profile. The increase of wave height also leads to increases of both first and second longshore current peaks, as shown in Table 2 by the values of two velocity peaks corresponding to different wave heights. This is easily understood for the first peak, as the maximum of longshore current being proportional to the magnitude of wave orbital velocity(v ðghb Þ1=2 γ =2, v is the mean longshore current velocity, γ wave breaking index representing saturation, hb water depth in breaking points) and the latter is proportional to wave height. The reason for the second peak is not so clear, as it is determined by the wave height at the second breaking point. It is seen in the figure that for increased wave height, the wave height at second breaking point is also increased and this results in the increased second velocity peak. But why the second peak increases with the increasing of incident wave height is not very clear, and this is because the process of second wave breaking is not understand very clearly up to now. The further study is needed for this problem. 4.3. Effect of wave period In order to present the effect of wave period on the longshore current distribution, the velocity profile for different wave periods (T ¼1.0 s, 1.5 s, 2.0 s) but similar wave height are plotted together in Fig. 7. Although the longshore uniformity of longshore current for long periods (T ¼ 1.5 s and 2.0 s) for slope 1:100 are not good, qualitative describing cross-shore distribution characteristics are acceptable. Table 3 summarizes the locations, values and the ratios of these two peaks for similar wave height but different period. It can be seen in the figure that the wave height from the slope toe to shoreline for long wave period was larger than that of short wave period, mainly near the offshore of the bar. This is because the change of wave heights caused by wave refraction depends on the wave period. For the similar incident wave heights, the wave height after wave refraction of long wave period is bigger than that of short wave period. It is seen from these results that the ratios of the first and second peaks is increased from smaller wave period (T¼ 1.0 s) to larger wave period (T ¼2.0 s) but their locations change little (x=xbar ¼ 0:90 and 0.25 for slope 1:40 and x=xbar ¼ 0:93 and 0.21 for slope 1:100). The later trend is the same as the cases when wave height is increased but wave period is fixed as shown in Fig. 6 and Table 2, but the former trend is different (the ratio does not change obviously when wave height is increased but wave period is fixed). The change of the ratio in Table 3 is caused mainly by the change of the values of second velocity peak with different wave period: the first peak change little for different wave period (but similar wave height). It is seen in Fig. 7 that the shapes of
velocity profiles are similar for different wave periods, but the larger wave period leads to onshore broadening of velocity profile at the first peak. This is similar to the result for the steep beach (1:20) given by Reniers and Battjes (1997). Another effect of increased wave period is that of increasing the wave height offshore of breaking point. 4.4. Effect of bottom topography The cross-shore profiles of longshore currents of slope 1:100 are wider than that of slope 1:40. This is because, for the 1:100 slope, the water depth is small and waves will begin to break early in its propagation on the slope, producing a wider surf zone and then a wider area covered by longshore currents (the lengths of the two slopes of 1:100 and 1:40 were identical in the experiment). The locations of first peak of longshore currents are all on onshore side of the bar. The locations of second peak of longshore currents of slope 1:100 are closer to the shoreline than that of slope 1:40. The value of second peak of slope 1:100 is smaller than that of slope 1:40. This is because, for the slope 1:40, the wave height after wave propagating over the bar is bigger.
5. Numerical modeling In order to investigate the formation process of the above bimodal distribution featrue of longshore current, numerical simulations were performed for the experimental conditions. The effects of lateral mixing, surface rollers and bottom friction on simulation results were discussed, which can show the physical process controlling the longshore current. The Nearshore POM model developed by Newberger and Allen (2007a, 2007b) has been extended to include wave-averaged forcing by breaking waves was adopted. The Nearshore POM circulation model is embedded within the NearCom community model and is coupled with a wave model. A wave model is required to provide the wave-averaged wave energy density, dissipation rate and wave number needed to force the waveaveraged circulation. 5.1. Hydrodynamic model The Nearshore POM has been widely used by thousands of users in all continents for a large variety of different applications. It is desirable in terms of computer economy to separate the vertically integrated equations (external mode) from the vertical structure equations (internal mode). The external mode of the POM, hydrodynamic model employed for the numeric simulations carried out in this work, will be presented: ∂η ∂UD ∂V D þ ¼0 þ ∂x ∂y ∂t ∂UD ∂U 2 D ∂UVD ∂η þ þ f V D þ gD F~x ¼ τsx τbx þ Gx ∂t ∂x ∂y ∂x ∂Sxx ∂Sxy ∂Sxxr ∂Sxyr þ þ þ ∂x ∂y ∂x ∂y ∂VD ∂UVD ∂V 2 D ∂η þ þ þ f UD þ gD F~y ¼ τsy τby þ Gy ∂t ∂x ∂y ∂y ∂Sxy ∂Syy ∂Sxyr ∂Syyr þ þ þ ∂x ∂y ∂x ∂y
ð1Þ
ð2Þ
ð3Þ
where x and y are horizontal coordinates, respectively;U and V are depth integrated velocity for x and y directions, respectively; t is time; g is gravity; η is the free surface; D is the total water depth, D ¼ η þ h. F~x and F~y are given by
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Fig. 7. Cross-shore profiles of wave height, setup and longshore current velocity of different wave periods. Left: tests 5, 9, 10; right: tests 1, 4, 5.
∂ ∂U ∂ ∂U ∂V D2AM þ DAM þ F~x ¼ ∂x ∂x ∂y ∂y ∂x
ð4Þ
∂ ∂V ∂ ∂U ∂V D2AM þ DAM þ F~y ¼ ∂y ∂y ∂x ∂y ∂x
ð5Þ
where AM is the horizontal eddy viscosity. Longuet-Higgins (1970) argued that the turbulent eddy viscosity should be a product of representative length and velocity and proposed the following formula: pffiffiffiffiffiffi AM ¼ Nx gh ð6Þ where N is a constant in the range of 0–0.016. According to experimental and numerical results (Zou et al., 2003) of plane beaches, its value used in this study is also set as 0.003. The terms designated Gx and Gy represent the small horizontal mixing, Gx ¼ Gy ¼ 0. The bottom shear stress is modeled 4
2
τbx ¼ ρf w U m U; τby ¼ ρf w U m V π π
ð7Þ
where U m ¼ Aω= sin hðkhÞ, f w is bottom friction coefficient,which is set 0.015 for two slopes.
5.2. Wave model We model this variation by assuming a narrow banded incident spectrum with Rayleigh distributed wave heights and utilizing the time dependent energy equation for the incident waves
(Özkan-Haller and Li, 2003): ∂E ∂ðEcg cos θÞ ∂ðEcg sin θÞ þ ¼ εd ∂x ∂y ∂t
ð8Þ
Here the parameter εd represents the wave-averaged breaking dissipation and is modeled (Church and Thornton, 1993): 2 3 pffiffiffiffi 3 ! 5=2 3 π B fp 3 H rms H rms 2 4 5 g H 1 1 1þ εd ¼ 1 þ tan h 8 16 D rms γD γD
ð9Þ where f p is the peak frequency of a narrow banded spectrum with assumed Rayleigh distributed wave heights; H rms represents the root mean square wave height; γ wave breaking index representing saturation, γ ¼ 0:7 in this paper; the coefficients used for the wave height transformation model is B ¼0.78. The parameters Sxx , Sxy and Syy in Eqs. (2) and (3) denote the components of the radiation stress tensor and are computed using linear water wave theory (Özkan-Haller and Kirby, 1999). The next equation is used to compute the roller energy Er : ∂Er ∂ ∂ ð2cEr cosθÞ þ ðcEr sinθÞ ¼ εd εr ∂y ∂t ∂x
ð10Þ
where c is the wave phase speed; the wave-averaged breaking dissipation εr is given by Duncan (1981):
εr ¼
g sin βEr c
ð11Þ
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where β represents the slope of the wave front,the empirical formula (Walstra et al., 1996): h H o 0:1 ð12Þ β ¼ 0:1kh H According to in Reniers's experiment (β ¼ 0:1 for slope 1:20), β ¼ 0:08 for slope 1:40 and β ¼ 0:03 for slope 1:100 are chosen in present paper. The radiation stress tensor of wave roller in Eqs. (2) and (3) are given as follows: sxxr ¼ 2Er cos 2 α; sxyr ¼ 2Er sin α cos α; syyr ¼ 2Er sin α 2
ð13Þ
As regards for boundaries conditions, the cross-shore boundary plays a key role in simulating longshore currents. A periodic boundary condition is imposed. Though lack of apparent physical explanation, the periodic condition has been successfully implemented in some previous research and it is the relatively simple settings for the cases studies in the present paper. 5.3. Comparison with measurements To compare the model results with the measurements, two tests have been selected: tests 1 and 7 for slopes 1:40 and 1:100 respectively. Fig. 8 shows the roller effect by comparing the numerical results of wave height H and mean water level η. It can be seen in the figures that wave height is well predicted and so is the setup if the roller effect is included. In the case where the roller is not included, the setup level between two wave breaking points is underestimated. For the examination of the effects of roller, bottom shear stress and lateral mixing on mean longshore current, tests 1 and 7 are chosen as an example, which shows the similar results for the other tests. The measurements for the longshore current velocity were performed at one third of the water depth from the bed, which in the case of a logarithmic velocity profile is representative for the depth-averaged flow. Fig. 9 presents the longshore current velocities computed with different values of lateral mixing coefficient N (N ¼0, 0.003), bottom friction coefficient, cf (cf ¼ 0:015; 0:012), roller slope coefficent β (β ¼0.05,0.08 for slope 1:40 and β ¼0.03,0.05 for slope 1:100). The cross-shore distribution of the longshore current velocity profile matches the measured distribution quite well. In particular, the location and the value of the computed and measured two peaks of longshore current velocity coincide. In addition to the location of second peak of slope 1:40 is underestimated (Fig. 9). This shows the
ability to predict the longshore current velocity profiles for purely wave driven flows on barred beaches, using existing model equations for longshore uniform conditions. The left panel of Fig. 9 shows that lateral mixing has no significant impact on the locations of two velocity peaks but can smooth longshore current velocity profile. The reason for the presence of the lateral mixing is because that there is concentrated wave breaking on the bar followed by a rapid decrease in the number of breaking waves in the trough, resulting in stronger shears and thereby increased mixing. The middle panel of Fig. 9 shows that decreasing the bottom friction coefficient cf produces an increase of longshore currents across the whole domain without shifting the locations of two velocity peaks. The right panel of Fig. 9 shows that the roller's effect is to modify the location of two velocity peaks of longshore currents. Excluding the roller dynamics (β ¼0), narrow current peaks were located on the seaward side of the shore-parallel bars (seen the dash line in right panel of Fig. 9). Including the roller dynamics (β ¼0.03,0.06) shifted the current maxima onshore and obviously increases the ratio of two peaks(the value of first peak decreasing and the value of second peak increasing), which effect increases with β decreasing. The above effects of N and cf are similar to those for steep beach (1:20) discussed by Reniers and Battjes (1997), but the effect of β is a bit different from that of Reniers and Battjes (1997): the increasing β (decreasing roller effect) leads to onshore shift of two peak and increasing the ratio of two peaks (the value of first peak decreasing and the value of second peak increasing).
6. Conclusions A comprehensive set of data, consisting of current velocity measurements on barred beaches with slopes of 1:100 and 1:40, as well as wave transformation and setup data, has been established. The data can be used for verification and validation of numerical models. The longshore current on barred beaches with two mild slopes (1:100 and 1:40) is studied experimentally, with focus on the effects of wave height, wave period and beach slope on bimodal feature of velocity distribution for the mild slope cases. It is found that the locations of two peaks of longshore current velocity remain unchanged for different wave heights, wave types and wave periods. The values of two velocity peaks increase with increasing wave height and period. The ratio of second peak to
Fig. 8. Wave height and setup of mean water level comparison between numerical results (solid and dash lines) and measurements ( ○△).
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Fig. 9. Comparison of computed longshore current velocities results (lines) and measurements ( ○△). The effects of (left) wave roller, (middle) bottom shear stress and (right) mixing are shown.
first peak is not changed for changing wave height (with a fixed wave period, T¼ 1.0 s), and increased for increasing wave period (from 1.0 s to 2.0 s). The corresponding numerical simulations are also performed with a simply model which is applicable for long straight beach. The effects of model parameters are examined and it is found that the lateral mixing has no significant impact on the locations of two velocity peaks but smoothes longshore current velocity profile, decreasing the bottom friction coefficient,cf , produces an increase of the whole velocity profile but does not shifting the locations of two velocity peaks. The roller has the effect of shifting the velocity profile shoreward and increases the values of two velocity peaks. Comparisons between numerical results and experimental data show that the agreement is good. Acknowledgments Project supported by the National Natural Science Foundation of China (Grant nos. 50479053 and 11272078) and the Science Fund for Creative Research Groups of China (51221961). References Battjes, J.A. and Janssen, J.P.F.M., 1978. Energy toss and set-up due to breaking in random waves. In: Proc. 16th Coastal Eng. Conf., Hamburg, pp. 569–587. Choi, J, Yoon, S.B., 2011. Numerical simulation of nearshore circulation on field topography in a random wave environment. Coast. Eng. 58, 395–408. Choi, J., Lee, J.I., Yoon, S.B., 2012. Surface roller modeling for mean longshore current over a barred beach in a random wave environment. J. Coast. Res. 28 (5), 1100–1120. Church, C.C., Thornton, E.B., 1993. Effects of breaking wave induced turbulence within a longshore current model. Coast. Eng. 20, l–28. Dalrymple, R.A., Birkemeier, W.A., Eubanks, R.A., 1977. Wave-induced circulation in shallow basins. J. Waterw. Port Coast. Ocean Div. 103 (1), 117–135.
Feddersen, F., Guza, R., 2003. Observation of nearshore circulation: alongshore uniformity. J. Geophys. Res. 108 (C3), 3006. Goda, Y., 2006. Examination of the influence of several factors on longshore current computation with random waves. Coast. Eng. 53, 157–170. Duncan, J.H., 1981. An Experimental Investigation of Breaking Waves Produced By a Towed Hydrofoil. Proceedings of the Royal Society of London 377, 331–348. Longuet-Higgins, M.S., 1970. Longshore currents generated by obliquely incident sea waves. J. Geophys. Res. 75, 6778–6801. Nayak, S, Bhaskaran, P K, Venkatesan, R., 2012. Near-shore wave-induced setup along Kalpakkam coast during an extreme cyclone event in the Bay of Bengal. Ocean Eng. 55, 52–61. Newberger, P.A., Allen, J.S., 2007a. Forcing a three-dimensional, hydrostatic, primitive-equation model for application in the surf zone: 1. Formulation. J. Geophys. Res. 112 (C08), 1–12. Newberger, P.A., Allen, J.S., 2007b. Forcing a three-dimensional, hydrostatic, primitive-equation model for application in the surf zone: 2. Application to DUCK94. J. Geophys. Res. 112 (C08), 1–21. Özkan-Haller, H.T., Li, Y., 2003. Effects of wave-current interaction on shear instabilities of longshore currents. J. Geophys. Res. 108 (C5), 3139. Özkan-Haller, H.T., Kirby, J.T., 1999. Nonlinear evolution of shear instabilities of the longshore current: a comparison of observations and computations. J. Geophys. Res.: Oceans 104 (C11), 25953–25984. Ren, C.P., Zou, Z.L., Qiu, D.H., 2011. Experimental study of the instabilities of alongshore currents on plane beaches. Coast. Eng. 59 (1), 72–89. Reniers, A.J.H.M., Battjes, J.A., 1997. A laboratory study of longshore currents over barred and non-barred beaches. Coast. Eng. 30, 1–21. Ruessink, B., Miles, J., Feddersen, F., Guza, R., Elgar, S., 2001. Modeling the alongshore current on barred beaches. J. Geophys. Res. 106, 22451–22463. Svendsen, L.A., Putrevu, U., 1994. Nearshore mixing and dispersion. Proc. R. Soc. Lond. A 445, 561–576. Tajima, Y., Madsen, O.S., 2006. Modeling near-shore waves, surface rollers and undertow velocity profiles. J.Waterw., Port., Coast.and Ocean Eng. ASCE 132 (6), 429–438. Visser P.J. 1984. Uniform longshore current measurements and calculations. In: Proceedings of the 19th International Conference on Coastal Engineering. Houston, TX, pp. 2192–2207. Walstra, D.J.R., Mocke, G.P., Smit, F., 1996. Roller contributions as inferred from inverse modelling techniques. Coast. Eng. Proc. 1, 25. Zou, Z.L., Wang, S.P., Qiu, D.H., Wang, Y., Wang, F.L, Dong, G.H., 2003. Longshore currents of regular waves on different beaches. Acta Oceanol. Sin. 22 (1), 123–132.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
An experimental and numerical study of bimodal velocity profile of longshore currents over mild-slope barred beaches Yan Wang n, Zhili Zou State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, China
art ic l e i nf o
a b s t r a c t
Article history: Received 24 October 2014 Accepted 24 June 2015
A detailed description is given of the results of laboratory experiments on bimodal velocity profile of longshore currents over two mild-slope barred beaches. The objective is to examine the bimodal longshore current velocity profile for purely wave-driven currents, with emphasis on the second peak and ratio of two peaks. The research focus is on the locations, values and ratios of two velocity peaks for the cases of mild slope beaches. The dependence of its bimodal feature on wave height, wave period and beach slope on the longshore currents are discussed. The 2-D model based on vertically integrated equations (external mode) of Nearshore POM was performed to compute the measured velocity profile. The depth-varying wave-induced residual momentum, the surface rollers, and the lateral mixing the bottom friction are considered, and the effects of lateral mixing, surface rollers and bottom friction on numerical model are discussed. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Mild slope Barred beach Bimodal feature Second peak POM model Wave roller
1. Introduction When water wave on beach propagates obliquely to the shoreline and break, a mean current is generated flowing parallel to the coast. For alongshore uniform barred beaches, the distribution of longshore current usually has two peaks, each corresponding to one of wave breaking points. The maximum peak of longshore current is expected to occur at the point of most intense wave breaking (Ruessink et al.,2001; Feddersen and Guza, 2003), i.e. on the bar crest. The minimum peak is expected to be close to the shore line. But in the case of field measurement, it is shown that a maximum longshore current velocity happens in the bar trough, instead of bar crest, as observed during the DELILAH field experiment results at Duck, North Carolina and the PNEC field experiment results on French Aquitanian coast beaches. Laboratory experiments on the shear instability of longshore currents have been carried out (Reniers and Battjes, 1997; Ren et al., 2011). This is contrary to the corresponding laboratory experiments and numerical simulation (Reniers and Battjes, 1997; Ruessink et al., 2001), which give the maximum velocity peak near the bar crest. Several mechanisms have been examined to explain this difference, including wave breaking turbulence, rollers, shear instabilities and alongshore pressure gradients (Putrevu and Svendsen, 1994). But these studies are most for relatively steep beach. The present study examines this problem for a mild slope case.
n
Corresponding author. E-mail address: [email protected] (Y. Wang).
http://dx.doi.org/10.1016/j.oceaneng.2015.06.038 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
The related experiment study for steep beach (1:20) was carried out (Reniers and Battjes, 1997). They used an external circulation system to maintain the flow circulation on beach caused by longshore current, which is similar to that of Visser (1984) for the longshore current on a plane beach. Their results show that the maximum velocity peak is on the bar crest. This result is also confirmed by their numerical simulation. The second velocity peak is not examined in detail in this study, because not enough velocity meters were adopted, and the exact location of velocity peak cannot be measured. Goda (2006) and Ruessink et al. (2001) also performed similar numerical simulations to investigate the cross-shore longshore current distributions on barred beach and similar results were obtained. Choi and Yoon (2011) and Choi et al. (2012) improved the wave-breaking model of Battjes and Janssen (1978) and the surface roller model of Tajima and Madsen (2006) for the application of SWAN-SHORECRC into the Sandy Duck experiments and its application showed satisfactory performances. Newberger and Allen (2007a, 2007b) added wave forcing in the form of surface stress and body forces in the Princeton Ocean Model (POM), which has evolved as “Nearshore POM”. Nayak et al. (2012) simulated the nearshore wave-induced setup along Kalpakkam coast during an extreme cyclone event in the Bay of Bengal using the SWAN wave model and the vertically integrated momentum balance equation in which a wave-induced setup is balanced by the wave radiation stresses. The present study investigates the above problem experimentally for the two mild slope cases and more velocity meters are adopt in all to measure the velocity peak, especial minimum peak.
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The numerical simulations are also conducted to explain the measurement. The purpose of the present paper is to investigate the bimodal longshore current velocity profile for purely wavedriven currents, with emphasis on the second peak and ratio of two peaks. In the present study, the longshore currents produced by regular waves are measured on the two barred beaches with slopes of 1:100 and 1:40. In view of the discussion on the position of the two peaks of longshore current in relation to the areas where wave energy is being dissipated by two wave breaking, we feel that the measurements on the cross-shore distribution of the longshore current velocity are valuable enough to be presented separately. The results can be used to obtain a better understanding of the forcing of wave-driven currents in the field. Experiment techniques of passive recirculation system are studied. The computational methods of longshore currents are investigated for simulation of the experimental results of longshore currents and wave heights for regular wave cases. The organization of the paper is as follows. Following the introduction, Section 2 presents the experiment setup. Section 3 shows the alongshore uniformity. Section 4 discusses the bimodal features of longshore currents. Section 5 establishes the numerical model with the effects of lateral mixing, bottom friction and surface rollers to compute the measured velocity profile and discusses the effect of lateral mixing, surface rollers and bottom shear stress on numerical results.
Fig. 1. Experimental setup (top) and layout of ADV (bottom). Measurement in meters.
2. Experiment setup The measurements were performed in the wave basin of the State Key Laboratory of Coastal and Offshore Engineering in Dalian University of Technology, which is 55 m long, 34 m wide and 0.7 m deep. Fig. 1 shows the experimental setup. In order to examine the bimodal feature of longshore current over bared beach with mild slope, 1:100 and 1:40 slope beaches were made by concrete. In order to form an obliquely incident wave on the beach and increase the beach length, the beach was rotated at 30o with respect to the wave-maker, which is at the one end of the wave basin. At each end of the beach a channel was made to maintain water circulation caused by longshore currents. The channels around the beach topography had the same depth as that in the flat part in front of the beach. Two wave guide walls were conducted on the side boundaries on the horizontal bottom part of the flow region. The wave absorbing layers formed by net boxes filled with plastic scraps were placed on the inner side of to prevent the wave reflection against the walls. Fig. 2 shows experimental photo of slope 1:100. The coordinate system (x, y) with the origin at the upstream end of still water shoreline was adopted with x-axis directing offshore and y-axis downstream. The measurements were performed on two barred concrete mild slopes with 1:40 and 1:100 slopes (Fig. 1) on which a Gaussian bar profile with a crest height of about 0.08 m or 0.04 m, width of 2.0 m and located at x¼ 5.0 m or 7.0 m was superimposed. The Gaussian bar profile was selected to reflect conditions observed in nature, with intense wave breaking over the bar, followed by waves reforming in the trough and additional wave breaking near the shoreline. The wave surface elevations were measured by 60 capacitance wave gauges placed in three measuring sections (I–III), which were normal to shoreline and have equal interval of 5 m. The distance from the upper end of the beach to measuring section I is y¼7 m. The measuring sections I and III both have 14 gauges with interval 2 m. The measuring section II has more wave gauges (32) and small gauge interval (0.5 m from 0.5 m to 10 m and 1 m from 10 m to 22 m) in order to measure the location of breaking point more accurately. The measurement results of these three
Fig. 2. Photograph of the experiment.
measuring lines were used to check the longshore uniformity of wave height, wave setup and set down. A total of 28 velocity meters (ADVs) were used for velocity measurement. They are divided into two sets, one being arranged in the longshore direction to measure the uniformity of longshore current (located at x ¼4 m (slope 1:100) and x ¼3 m (slope 1:40) with interval 2 m); the other being arranged in the cross-shore direction to measure the cross-shore distributions (located at y¼14.5 m intervals 0.5–2.0 m). The vertical distance from bottom of each ADV is 1/3 local water depth for the measurement of depth-average velocity. The sample length of ADVs recording was 600 s. Fig. 3 shows time series of 5 flow meters in section IV in Fig. 1, from which we can see that the recording of every flow meter has a steady mean value after 200 s, indicating that the steady longshore current is achieved. We also see the fluctuation of the longshore current from the figure, and the period of fluctuation is much longer than the wave period (wave period is 1.0 s). This shows that the flow meters cannot respond to wave velocity simultaneously and only give the results of wave averaged velocity. The steady velocity data cut from about 300 s in the time series were used to determine the mean longshore currents. The mean longshore velocity is obtained by time-averaging of the recorded velocity time series cut by about 300 s from the time series beginning. Unidirectional obliquely incident regular and irregular waves were generated by the multi-paddle-type wave maker. Table 1 lists
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the incident wave conditions, where T is the wave period, H 0 is mean wave height for regular wave and the significant wave height, h is the still water depth over horizontal bottom.
3. Alongshore uniformity Different from the previous researches, a passive, instead of active, circulation system is adopt to form flow circulation caused by the longshore currents, as the active circulation system (Reniers and Battjes, 1997) needs a manmade flow flux in the up and down stream ends of the beach which is difficult to determined accurately and will affect the accuracy of measured longshore current. The passive circulation system employed in the present study was formed by the channel around the beach model, as shown in Fig. 1 by arrows. The channel width is 4.4 m at the two sides of the beach and 4.0–8.0 m behind the beach. The channel water depth is the same as that in the horizontal bottom in front of the beach. The flow in the downstream channel will be driven by the longshore currents and goes through the channel behind the beach model to reach the upstream channel. Then, it feeds into the upstream end to form a closed flow circulation. The longshore current formed by this circulation system has a small longshore velocity at the upstream end (because the driving force (radiation
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stress) does not get properly formed due to the lateral boundary effect) and grows linearly to almost uniformed longshore velocity over a milder part of the beach, and then have gradual increase toward the downstream end (because of the same reason as at the upstream end). This form of a passive circulation system is similar to that discussed theoretically and experimentally by Dalrymple et al. (1977), which has a closed upstream end but an open downstream end. The theoretical and experimental streamline of such a system were given and it is seen that uniformly distributed streamline are formed over the middle and downstream part of the beach. The similar longshore flow pattern was also found in the present experiment, as shown in Fig. 4. This is especially true for a short wave period case (T¼ 1.0 s) (see the discussion of following subsection). The longshore uniformity of longshore currents generated by the passive circulations system is examined by the mean longshore current recorded by the alongshore set of ADVs. Fig. 4 shows such longshore currents of different wave height for each wave period considered, in which results along the measuring line, x¼3 m for slope 1:40 and x¼ 4 m for slope 1:100 are presented. Another wave period (1.5 s and 2.0 s) for each of the last 4 panels in the figure (which is not listed in Table 1 as it is only considered here) is added in order to consider the wave conditions for three wave periods. It can be seen for the cases with T¼1.0 s and slopes 1:40, 1:100, and the cases with T¼1.5 s and slope 1:40 that the longshore currents is uniform from y¼8.5 m to 18.5 m, and the longshore currents velocity in the upstream is smaller and the longshore currents velocity in the downstream is larger. For the cases with T¼ 2.0 s and slope 1:40, the longshore currents velocity becomes significantly increased from y¼ 12.5 m to 22.5 m and the region of longshore uniformity gets shorter. For the cases with T¼1.5 s or 2.0 s and slope 1:100, the apparent fluctuations are shown in longshore direction. Therefore it was found that, with the increased wave period, the longshore uniformity of longshore current get worse and the region of uniform longshore current becoming shorter. The cases with T¼1.0 s and slopes 1:40, 1:100, and the cases with T¼ 1.5 s, 2.0 s and slope 1:40, the length of the uniform longshore current is long (one third of total beach length) enough for the measurement of the longshore current velocity profile. The longshore uniformity of the wave height and wave setup was also checked by the surface elevation measurement in measuring sections I, II and III. Fig. 5 shows the measured wave height and setup for test 1 (slope 1:40) and test 7 (slope 1:100) over barred beaches. As shown in figure, the wave height and wave setup are approximately uniform in the basin for barred beaches, which form mean longshore current. This is important in view of the question raised in Section 1, since it implies absence of an alongshore pressure gradient and a purely local wave-induced driving force. Three runs for each test condition were conducted and it is found that the repeatability of these results is quite good.
4. Bimodal features of cross-shore distribution of longshore currents In this section the factors influencing the cross-shore distribution of measured longshore currents are presented, which include the wave height, wave period and bottom topography, whose
Fig. 3. The time series of longshore current velocity (test 2). Table 1 Wave conditions. Test
1
2
3
4
5
6
7
8
9
10
Slope d/m T/s H 0 /m
1:40 0.45 1.0 0.070
1:40 0.45 1.0 0.090
1:40 0.45 1.0 0.12
1:40 0.45 1.5 0.070
1:40 0.45 2.0 0.070
1:100 0.18 1.0 0.025
1:100 0.18 1.0 0.050
1:100 0.18 1.0 0.060
1:100 0.18 1.5 0.050
1:100 0.18 2.0 0.050
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Fig. 4. A longshore distribution of mean longshore current.
Fig. 5. Longshore uniformity of wave height and setup for barred beaches (tests 1, 7).
effect is given in details in the next subsection. Fig. 6 shows the cross-shore profiles of wave height, setup and longshore current velocity, which correspond to different wave heights of T ¼1.0 s shown in Table 1. The measurement data of wave height and setup is from the measuring section II in Fig. 1. The left panel is for the cases of slope 1:100 and the right panel for the cases of slope 1:40.
The main feature of the velocity profile which is different from the plane beach is bimodal: there are two velocity peaks corresponding to two wave breakings. Table 2 summarizes the locations, values and the ratios of these two peaks for same period (1.0 s) but different wave height. The following gives the details discussion of the results.
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Fig. 6. Cross-shore profiles of wave height, setup and longshore current velocity of T ¼1.0 s.
Table 2 First and second velocity peaks for same period (1.0 s) but different wave height. Test xð1Þ =xbar xð2Þ =xbar First peak b b Value/ m s1
Location
xð1Þ c =xbar
xð2Þ c =xbar
Value/ m s1
0.90 0.90 0.90 0.93 0.93 0.93
0.276 0.369 0.421 0.067 0.146 0.153
0.25 0.25 0.25 0.21 0.21 0.21
0.116 0.153 0.169 0.018 0.040 0.044
Location
1 2 3 6 7 8
1.1 1.3 1.6 1.07 1.43 1.71
0.30 0.30 0.40 0.29 0.29 0.29
Second peak
Ratio of two peaks
0.42 0.41 0.40 0.27 0.27 0.28
ð2Þ xð1Þ ; xð2Þ : locations of two wave breaking points; xð1Þ c ; xc : locations of two peaks of b b longshore currents.
4.1. Locations and values of two peaks It is seen from the wave height distributions in the figures that with wave propagating toward shoreline the wave breaking happens two times for the barred beaches. The first wave breaking occur mostly on offshore sides of sand bar, because the water depth around bar crest is 0.03 m (slope 1:100) and 0.045 m (slope 1:40), which is so small that the breaking condition H 4 γ h can be satisfied easily on offshore sides of sand bar. The second wave breaking occurs near the shoreline. Corresponding, there are two peaks of longshore current velocity profile. As the first peak of longshore current corresponds to the first wave breaking, it is controlled by the sand bar. But there is a distance lag between
velocity peak and breaking point, as shown in Table 2 by the values of xbð1Þ =xbar and xð1Þ c =xbar : the difference between two values is the distance lag. For the second velocity peak and second wave breaking point, this lag also exist, which is shown by the values of xð2Þ =xbar and xð2Þ c =xbar in Table 2. These results show that the first b peak of longshore current velocity all occurs at the onshore side of ð1Þ sand bar: xð1Þ c =xbar ¼ 0:90 for slope 1:40 and xc =xbar ¼ 0:93 for slope 1:100. This result is a little bit difference from the experimental result (xcð1Þ =xbar ¼ 1:0) by Reniers and Battjes (1997). This may be because there was not measurement points between the crest and trough of the bar in their experiment, so the exact location of velocity peak cannot be determined for their experiment. Both the present and Reniers and Battjes's results are different from the field experiments of DELILAH (1991) and PNEC (2006), which indicate that the first peak of longshore current velocity is located in the bar trough. The second peak of longshore current all occurs near the shoreline, x¼1.5 m (corresponding to ð2Þ xð2Þ c =xbar ¼ 0:21) and x ¼1.25 m (corresponding to xc =xbar ¼ 0:25), where the remaining wave energy was dissipated after propagating over the bar. Table 3 gives the ratio of second peak to first peak for different cases, which are near 0.41 (slope 1:40) and 0.27 (slope 1:100). It is interesting to notice that for each slope, the locations and ratio of the two peaks in the table remain unchanged for different wave heights and wave types. 4.2. Effect of wave height The wave height effect is studied in the present experiment by varying the wave heights of incident wave for a fixed wave period,
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Table 3 First and second velocity peaks for similar wave height but different period Test T/s First peak
1.0 1.5 2.0 1.0 1.5 2.0
Ratio of two Figure peaks
Value/ m s1
Location
xð1Þ c =xbar
xð2Þ c =xbar
Value/ m s1
0.90 0.90 0.90 0.93 0.93 0.93
0.276 0.275 0.301 0.067 0.151 0.152
0.25 0.25 0.25 0.21 0.21 0.21
0.42 0.55 0.59 0.27 0.31 0.39
Location
1 4 5 6 9 10
Second peak
0.116 0.151 0.179 0.018 0.046 0.059
Fig. 7 (a)–(c) Fig. 7 (d)–(f)
ð2Þ xð1Þ c ; xc : locations of two peaks of longshore currents.
which are from 0.07 m to 0.12 cm for slope 1:40 and 0.025 m to 0.060 cm for slope 1:100. It is seen in Fig. 4 that the increasing of wave height results in wave breaking starting further offshore, and this leads to a broader breaking zone and a broader current velocity profile. The increase of wave height also leads to increases of both first and second longshore current peaks, as shown in Table 2 by the values of two velocity peaks corresponding to different wave heights. This is easily understood for the first peak, as the maximum of longshore current being proportional to the magnitude of wave orbital velocity(v ðghb Þ1=2 γ =2, v is the mean longshore current velocity, γ wave breaking index representing saturation, hb water depth in breaking points) and the latter is proportional to wave height. The reason for the second peak is not so clear, as it is determined by the wave height at the second breaking point. It is seen in the figure that for increased wave height, the wave height at second breaking point is also increased and this results in the increased second velocity peak. But why the second peak increases with the increasing of incident wave height is not very clear, and this is because the process of second wave breaking is not understand very clearly up to now. The further study is needed for this problem. 4.3. Effect of wave period In order to present the effect of wave period on the longshore current distribution, the velocity profile for different wave periods (T ¼1.0 s, 1.5 s, 2.0 s) but similar wave height are plotted together in Fig. 7. Although the longshore uniformity of longshore current for long periods (T ¼ 1.5 s and 2.0 s) for slope 1:100 are not good, qualitative describing cross-shore distribution characteristics are acceptable. Table 3 summarizes the locations, values and the ratios of these two peaks for similar wave height but different period. It can be seen in the figure that the wave height from the slope toe to shoreline for long wave period was larger than that of short wave period, mainly near the offshore of the bar. This is because the change of wave heights caused by wave refraction depends on the wave period. For the similar incident wave heights, the wave height after wave refraction of long wave period is bigger than that of short wave period. It is seen from these results that the ratios of the first and second peaks is increased from smaller wave period (T¼ 1.0 s) to larger wave period (T ¼2.0 s) but their locations change little (x=xbar ¼ 0:90 and 0.25 for slope 1:40 and x=xbar ¼ 0:93 and 0.21 for slope 1:100). The later trend is the same as the cases when wave height is increased but wave period is fixed as shown in Fig. 6 and Table 2, but the former trend is different (the ratio does not change obviously when wave height is increased but wave period is fixed). The change of the ratio in Table 3 is caused mainly by the change of the values of second velocity peak with different wave period: the first peak change little for different wave period (but similar wave height). It is seen in Fig. 7 that the shapes of
velocity profiles are similar for different wave periods, but the larger wave period leads to onshore broadening of velocity profile at the first peak. This is similar to the result for the steep beach (1:20) given by Reniers and Battjes (1997). Another effect of increased wave period is that of increasing the wave height offshore of breaking point. 4.4. Effect of bottom topography The cross-shore profiles of longshore currents of slope 1:100 are wider than that of slope 1:40. This is because, for the 1:100 slope, the water depth is small and waves will begin to break early in its propagation on the slope, producing a wider surf zone and then a wider area covered by longshore currents (the lengths of the two slopes of 1:100 and 1:40 were identical in the experiment). The locations of first peak of longshore currents are all on onshore side of the bar. The locations of second peak of longshore currents of slope 1:100 are closer to the shoreline than that of slope 1:40. The value of second peak of slope 1:100 is smaller than that of slope 1:40. This is because, for the slope 1:40, the wave height after wave propagating over the bar is bigger.
5. Numerical modeling In order to investigate the formation process of the above bimodal distribution featrue of longshore current, numerical simulations were performed for the experimental conditions. The effects of lateral mixing, surface rollers and bottom friction on simulation results were discussed, which can show the physical process controlling the longshore current. The Nearshore POM model developed by Newberger and Allen (2007a, 2007b) has been extended to include wave-averaged forcing by breaking waves was adopted. The Nearshore POM circulation model is embedded within the NearCom community model and is coupled with a wave model. A wave model is required to provide the wave-averaged wave energy density, dissipation rate and wave number needed to force the waveaveraged circulation. 5.1. Hydrodynamic model The Nearshore POM has been widely used by thousands of users in all continents for a large variety of different applications. It is desirable in terms of computer economy to separate the vertically integrated equations (external mode) from the vertical structure equations (internal mode). The external mode of the POM, hydrodynamic model employed for the numeric simulations carried out in this work, will be presented: ∂η ∂UD ∂V D þ ¼0 þ ∂x ∂y ∂t ∂UD ∂U 2 D ∂UVD ∂η þ þ f V D þ gD F~x ¼ τsx τbx þ Gx ∂t ∂x ∂y ∂x ∂Sxx ∂Sxy ∂Sxxr ∂Sxyr þ þ þ ∂x ∂y ∂x ∂y ∂VD ∂UVD ∂V 2 D ∂η þ þ þ f UD þ gD F~y ¼ τsy τby þ Gy ∂t ∂x ∂y ∂y ∂Sxy ∂Syy ∂Sxyr ∂Syyr þ þ þ ∂x ∂y ∂x ∂y
ð1Þ
ð2Þ
ð3Þ
where x and y are horizontal coordinates, respectively;U and V are depth integrated velocity for x and y directions, respectively; t is time; g is gravity; η is the free surface; D is the total water depth, D ¼ η þ h. F~x and F~y are given by
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Fig. 7. Cross-shore profiles of wave height, setup and longshore current velocity of different wave periods. Left: tests 5, 9, 10; right: tests 1, 4, 5.
∂ ∂U ∂ ∂U ∂V D2AM þ DAM þ F~x ¼ ∂x ∂x ∂y ∂y ∂x
ð4Þ
∂ ∂V ∂ ∂U ∂V D2AM þ DAM þ F~y ¼ ∂y ∂y ∂x ∂y ∂x
ð5Þ
where AM is the horizontal eddy viscosity. Longuet-Higgins (1970) argued that the turbulent eddy viscosity should be a product of representative length and velocity and proposed the following formula: pffiffiffiffiffiffi AM ¼ Nx gh ð6Þ where N is a constant in the range of 0–0.016. According to experimental and numerical results (Zou et al., 2003) of plane beaches, its value used in this study is also set as 0.003. The terms designated Gx and Gy represent the small horizontal mixing, Gx ¼ Gy ¼ 0. The bottom shear stress is modeled 4
2
τbx ¼ ρf w U m U; τby ¼ ρf w U m V π π
ð7Þ
where U m ¼ Aω= sin hðkhÞ, f w is bottom friction coefficient,which is set 0.015 for two slopes.
5.2. Wave model We model this variation by assuming a narrow banded incident spectrum with Rayleigh distributed wave heights and utilizing the time dependent energy equation for the incident waves
(Özkan-Haller and Li, 2003): ∂E ∂ðEcg cos θÞ ∂ðEcg sin θÞ þ ¼ εd ∂x ∂y ∂t
ð8Þ
Here the parameter εd represents the wave-averaged breaking dissipation and is modeled (Church and Thornton, 1993): 2 3 pffiffiffiffi 3 ! 5=2 3 π B fp 3 H rms H rms 2 4 5 g H 1 1 1þ εd ¼ 1 þ tan h 8 16 D rms γD γD
ð9Þ where f p is the peak frequency of a narrow banded spectrum with assumed Rayleigh distributed wave heights; H rms represents the root mean square wave height; γ wave breaking index representing saturation, γ ¼ 0:7 in this paper; the coefficients used for the wave height transformation model is B ¼0.78. The parameters Sxx , Sxy and Syy in Eqs. (2) and (3) denote the components of the radiation stress tensor and are computed using linear water wave theory (Özkan-Haller and Kirby, 1999). The next equation is used to compute the roller energy Er : ∂Er ∂ ∂ ð2cEr cosθÞ þ ðcEr sinθÞ ¼ εd εr ∂y ∂t ∂x
ð10Þ
where c is the wave phase speed; the wave-averaged breaking dissipation εr is given by Duncan (1981):
εr ¼
g sin βEr c
ð11Þ
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where β represents the slope of the wave front,the empirical formula (Walstra et al., 1996): h H o 0:1 ð12Þ β ¼ 0:1kh H According to in Reniers's experiment (β ¼ 0:1 for slope 1:20), β ¼ 0:08 for slope 1:40 and β ¼ 0:03 for slope 1:100 are chosen in present paper. The radiation stress tensor of wave roller in Eqs. (2) and (3) are given as follows: sxxr ¼ 2Er cos 2 α; sxyr ¼ 2Er sin α cos α; syyr ¼ 2Er sin α 2
ð13Þ
As regards for boundaries conditions, the cross-shore boundary plays a key role in simulating longshore currents. A periodic boundary condition is imposed. Though lack of apparent physical explanation, the periodic condition has been successfully implemented in some previous research and it is the relatively simple settings for the cases studies in the present paper. 5.3. Comparison with measurements To compare the model results with the measurements, two tests have been selected: tests 1 and 7 for slopes 1:40 and 1:100 respectively. Fig. 8 shows the roller effect by comparing the numerical results of wave height H and mean water level η. It can be seen in the figures that wave height is well predicted and so is the setup if the roller effect is included. In the case where the roller is not included, the setup level between two wave breaking points is underestimated. For the examination of the effects of roller, bottom shear stress and lateral mixing on mean longshore current, tests 1 and 7 are chosen as an example, which shows the similar results for the other tests. The measurements for the longshore current velocity were performed at one third of the water depth from the bed, which in the case of a logarithmic velocity profile is representative for the depth-averaged flow. Fig. 9 presents the longshore current velocities computed with different values of lateral mixing coefficient N (N ¼0, 0.003), bottom friction coefficient, cf (cf ¼ 0:015; 0:012), roller slope coefficent β (β ¼0.05,0.08 for slope 1:40 and β ¼0.03,0.05 for slope 1:100). The cross-shore distribution of the longshore current velocity profile matches the measured distribution quite well. In particular, the location and the value of the computed and measured two peaks of longshore current velocity coincide. In addition to the location of second peak of slope 1:40 is underestimated (Fig. 9). This shows the
ability to predict the longshore current velocity profiles for purely wave driven flows on barred beaches, using existing model equations for longshore uniform conditions. The left panel of Fig. 9 shows that lateral mixing has no significant impact on the locations of two velocity peaks but can smooth longshore current velocity profile. The reason for the presence of the lateral mixing is because that there is concentrated wave breaking on the bar followed by a rapid decrease in the number of breaking waves in the trough, resulting in stronger shears and thereby increased mixing. The middle panel of Fig. 9 shows that decreasing the bottom friction coefficient cf produces an increase of longshore currents across the whole domain without shifting the locations of two velocity peaks. The right panel of Fig. 9 shows that the roller's effect is to modify the location of two velocity peaks of longshore currents. Excluding the roller dynamics (β ¼0), narrow current peaks were located on the seaward side of the shore-parallel bars (seen the dash line in right panel of Fig. 9). Including the roller dynamics (β ¼0.03,0.06) shifted the current maxima onshore and obviously increases the ratio of two peaks(the value of first peak decreasing and the value of second peak increasing), which effect increases with β decreasing. The above effects of N and cf are similar to those for steep beach (1:20) discussed by Reniers and Battjes (1997), but the effect of β is a bit different from that of Reniers and Battjes (1997): the increasing β (decreasing roller effect) leads to onshore shift of two peak and increasing the ratio of two peaks (the value of first peak decreasing and the value of second peak increasing).
6. Conclusions A comprehensive set of data, consisting of current velocity measurements on barred beaches with slopes of 1:100 and 1:40, as well as wave transformation and setup data, has been established. The data can be used for verification and validation of numerical models. The longshore current on barred beaches with two mild slopes (1:100 and 1:40) is studied experimentally, with focus on the effects of wave height, wave period and beach slope on bimodal feature of velocity distribution for the mild slope cases. It is found that the locations of two peaks of longshore current velocity remain unchanged for different wave heights, wave types and wave periods. The values of two velocity peaks increase with increasing wave height and period. The ratio of second peak to
Fig. 8. Wave height and setup of mean water level comparison between numerical results (solid and dash lines) and measurements ( ○△).
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Fig. 9. Comparison of computed longshore current velocities results (lines) and measurements ( ○△). The effects of (left) wave roller, (middle) bottom shear stress and (right) mixing are shown.
first peak is not changed for changing wave height (with a fixed wave period, T¼ 1.0 s), and increased for increasing wave period (from 1.0 s to 2.0 s). The corresponding numerical simulations are also performed with a simply model which is applicable for long straight beach. The effects of model parameters are examined and it is found that the lateral mixing has no significant impact on the locations of two velocity peaks but smoothes longshore current velocity profile, decreasing the bottom friction coefficient,cf , produces an increase of the whole velocity profile but does not shifting the locations of two velocity peaks. The roller has the effect of shifting the velocity profile shoreward and increases the values of two velocity peaks. Comparisons between numerical results and experimental data show that the agreement is good. Acknowledgments Project supported by the National Natural Science Foundation of China (Grant nos. 50479053 and 11272078) and the Science Fund for Creative Research Groups of China (51221961). References Battjes, J.A. and Janssen, J.P.F.M., 1978. Energy toss and set-up due to breaking in random waves. In: Proc. 16th Coastal Eng. Conf., Hamburg, pp. 569–587. Choi, J, Yoon, S.B., 2011. Numerical simulation of nearshore circulation on field topography in a random wave environment. Coast. Eng. 58, 395–408. Choi, J., Lee, J.I., Yoon, S.B., 2012. Surface roller modeling for mean longshore current over a barred beach in a random wave environment. J. Coast. Res. 28 (5), 1100–1120. Church, C.C., Thornton, E.B., 1993. Effects of breaking wave induced turbulence within a longshore current model. Coast. Eng. 20, l–28. Dalrymple, R.A., Birkemeier, W.A., Eubanks, R.A., 1977. Wave-induced circulation in shallow basins. J. Waterw. Port Coast. Ocean Div. 103 (1), 117–135.
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